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PubMed2002

Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation.

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho) <or= 1, this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for lambda(1 - rho) close to 1(-) there is a family of periodic solutions similar to that in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small.

Gravitation↗
PubMed1991

Adaptive processing techniques based on hidden Markov models for characterizing very small channel currents buried in noise and deterministic interferences.

Techniques for characterizing very small single-channel currents buried in background noise are described and tested on simulated data to give confidence when applied to real data. Single channel currents are represented as a discrete-time, finite-state, homogeneous, Markov process, and the noise that obscures the signal is assumed to be white and Gaussian. The various signal model parameters, such as the Markov state levels and transition probabilities, are unknown. In addition to white Gaussian noise, the signal can be corrupted by deterministic interferences of known form but unknown parameters, such as the sinusoidal disturbance stemming from AC interference and a drift of the base line owing to a slow development of liquid-junction potentials. To characterize the signal buried in such stochastic and deterministic interferences, the problem is first formulated in the framework of a Hidden Markov Model and then the Expectation Maximization algorithm is applied to obtain the maximum likelihood estimates of the model parameters (state levels, transition probabilities), signals, and the parameters of the deterministic disturbances. Using fictitious channel currents embedded in the idealized noise, we first show that the signal processing technique is capable of characterizing the signal characteristics quite accurately even when the amplitude of currents is as small as 5-10 fA. The statistics of the signal estimated from the processing technique include the amplitude, mean open and closed duration, open-time and closed-time histograms, probability of dwell-time and the transition probability matrix. With a periodic interference composed, for example, of 50 Hz and 100 Hz components, or a linear drift of the baseline added to the segment containing channel currents and white noise, the parameters of the deterministic interference, such as the amplitude and phase of the sinusoidal wave, or the rate of linear drift, as well as all the relevant statistics of the signal, are accurately estimated with the algorithm we propose. Also, if the frequencies of the periodic interference are unknown, they can be accurately estimated. Finally, we provide a technique by which channel currents originating from the sum of two or more independent single channels are decomposed so that each process can be separately characterized. This process is also formulated as a Hidden Markov Model problem and solved by applying the Expectation Maximization algorithm. The scheme relies on the fact that the transition matrix of the summed Markov process can be construed as a tensor product of the transition matrices of individual processes.

Algorithms↗
PubMed2005

Anisotropy of the electron component in a cylindrical magnetron discharge. I. Theory of the multiterm analysis.

A general multiterm representation of the phase space electron distribution function in terms of spherical tensors is used to solve the Boltzmann kinetic equation in crossed electric and magnetic fields. The problem is formulated for an axisymmetric cylindrical magnetron discharge with the homogeneous magnetic field being directed axially and the electric field between the coaxial cathode and anode varying in radius only. A spherical harmonic representation of the velocity distribution function in Cartesian coordinates becomes especially cumbersome in the presence of the magnetic field. In contrast, the employment of a spherical tensor representation leads to a compact hierarchy of equations that accurately take into account the spatial inhomogeneities and anisotropy of the plasma in crossed fields. To describe the spatially inhomogeneous plasma the hierarchy of the kinetic equations is formulated in terms of the total energy and the radial coordinate. Appropriate boundary conditions at the electrodes for the tensor expansion coefficients are obtained.

Journal Article↗
PubMed2005

Predicting failure using conditioning on damage history: demonstration on percolation and hierarchical fiber bundles.

We formulate the problem of probabilistic predictions of global failure in the simplest possible model based on site percolation and on one of the simplest models of time-dependent rupture, a hierarchical fiber bundle model. We show that conditioning the predictions on the knowledge of the current degree of damage (occupancy density p or number and size of cracks) and on some information on the largest cluster improves significantly the prediction accuracy, in particular by allowing one to identify those realizations which have anomalously low or large clusters (cracks). We quantify the prediction gains using two measures, the relative specific information gain (which is the variation of entropy obtained by adding new information) and the root mean square of the prediction errors over a large ensemble of realizations. The bulk of our simulations have been obtained with the two-dimensional site percolation model on a lattice of size L x L=20 x 20 and hold true for other lattice sizes. For the hierarchical fiber bundle model, conditioning the measures of damage on the information of the location and size of the largest crack extends significantly the critical region and the prediction skills. These examples illustrate how ongoing damage can be used as a revelation of both the realization-dependent preexisting heterogeneity and the damage scenario undertaken by each specific sample.

Journal Article↗
PubMed2000

Adsorption on carbon nanotubes: quantum spin tubes, magnetization plateaus, and conformal symmetry.

We formulate the problem of adsorption onto the surface of a carbon nanotube as a lattice gas on a triangular lattice wrapped around a cylinder. This model is equivalent to an XXZ Heisenberg quantum spin tube. We find density plateau structures for armchair, zigzag, and chiral nanotubes. The zigzags are special and have extensive zero temperature entropy plateaus in the classical limit. Quantum effects lift the degeneracy, leaving gapless excitations described by a c = 1 conformal field theory with compactification radius quantized by the tube circumference.

Journal Article↗
PubMed2000

Bayesian analysis of blood glucose time series from diabetes home monitoring.

This paper describes the application of a novel Bayesian estimation technique to extract the structural components, i.e., trend and daily patterns, from blood glucose level time series coming from home monitoring of insulin dependent diabetes mellitus patients. The problem is formulated through a set of stochastic equations, and is solved in a Bayesian framework by using a Markov chain Monte Carlo technique. The potential of the method is illustrated by analyzing data coming from the home monitoring of a 14-year old male patient.

Adolescent↗
PubMed1999

Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms.

In order to perform attenuation correction in emission tomography an attenuation map is required. We propose a new method to compute this map directly from the emission sinogram, eliminating the transmission scan from the acquisition protocol. The problem is formulated as an optimization task where the objective function is a combination of the likelihood and an a priori probability. The latter uses a Gibbs prior distribution to encourage local smoothness and a multimodal distribution for the attenuation coefficients. Since the attenuation process is different in positron emission tomography (PET) and single photon emission tomography (SPECT), a separate algorithm for each case is derived. The method has been tested on mathematical phantoms and on a few clinical studies. For PET, good agreement was found between the images obtained with transmission measurements and those produced by the new algorithm in an abdominal study. For SPECT, promising simulation results have been obtained for nonhomogeneous attenuation due to the presence of the lungs.

Abdomen↗
PubMedDate not supplied

Significance of gene ranking for classification of microarray samples.

Many methods for classification and gene selection with microarray data have been developed. These methods usually give a ranking of genes. Evaluating the statistical significance of the gene ranking is important for understanding the results and for further biological investigations, but this question has not been well addressed for machine learning methods in existing works. Here, we address this problem by formulating it in the framework of hypothesis testing and propose a solution based on resampling. The proposed r-test methods convert gene ranking results into position p-values to evaluate the significance of genes. The methods are tested on three real microarray data sets and three simulation data sets with support vector machines as the method of classification and gene selection. The obtained position p-values help to determine the number of genes to be selected and enable scientists to analyze selection results by sophisticated multivariate methods under the same statistical inference paradigm as for simple hypothesis testing methods.

Algorithms↗
PubMed2004

Differentiation of discrete multidimensional signals.

We describe the design of finite-size linear-phase separable kernels for differentiation of discrete multidimensional signals. The problem is formulated as an optimization of the rotation-invariance of the gradient operator, which results in a simultaneous constraint on a set of one-dimensional low-pass prefilter and differentiator filters up to the desired order. We also develop extensions of this formulation to both higher dimensions and higher order directional derivatives. We develop a numerical procedure for optimizing the constraint, and demonstrate its use in constructing a set of example filters. The resulting filters are significantly more accurate than those commonly used in the image and multidimensional signal processing literature.

Algorithms↗
PubMed2004

Polygonal and polyhedral contour reconstruction in computed tomography.

This paper is about three-dimensional (3-D) reconstruction of a binary image from its X-ray tomographic data. We study the special case of a compact uniform polyhedron totally included in a uniform background and directly perform the polyhedral surface estimation. We formulate this problem as a nonlinear inverse problem using the Bayesian framework. Vertice estimation is done without using a voxel approximation of the 3-D image. It is based on the construction and optimization of a regularized criterion that accounts for surface smoothness. We investigate original deterministic local algorithms, based on the exact computation of the line projections, their update, and their derivatives with respect to the vertice coordinates. Results are first derived in the two-dimensional (2-D) case, which consists of reconstructing a 2-D object of deformable polygonal contour from its tomographic data. Then, we investigate the 3-D extension that requires technical adaptations. Simulation results illustrate the performance of polygonal and polyhedral reconstruction algorithms in terms of quality and computation time.

Algorithms↗
PubMed2006

Optimal requantization of deep grayscale images and Lloyd-Max quantization.

The classic signal quantization problem was introduced by Lloyd. We formulate another, similar problem: The optimal mapping of digital fine grayscale images (such as 9-13 bits-per-pixel medical images) to a coarser scale (e.g., 8 bits per pixel on conventional computer monitors). While the former problem is defined basically in the real signal domain with smoothly distributed noise, the latter refers to an essentially digital domain. As we show in this paper, it is this difference that makes the classic quantization methods virtually inapplicable in typical cases of requantization of the already digitized images. We found experimentally that an algorithm based on dynamic programming provides significantly better results than Lloyd's method.

Algorithms↗
PubMed2006

Concurrent 3-D motion segmentation and 3-D interpretation of temporal sequences of monocular images.

The purpose of this study is to investigate a variational method for joint multiregion three-dimensional (3-D) motion segmentation and 3-D interpretation of temporal sequences of monocular images. Interpretation consists of dense recovery of 3-D structure and motion from the image sequence spatiotemporal variations due to short-range image motion. The method is direct insomuch as it does not require prior computation of image motion. It allows movement of both viewing system and multiple independently moving objects. The problem is formulated following a variational statement with a functional containing three terms. One term measures the conformity of the interpretation within each region of 3-D motion segmentation to the image sequence spatiotemporal variations. The second term is of regularization of depth. The assumption that environmental objects are rigid accounts automatically for the regularity of 3-D motion within each region of segmentation. The third and last term is for the regularity of segmentation boundaries. Minimization of the functional follows the corresponding Euler-Lagrange equations. This results in iterated concurrent computation of 3-D motion segmentation by curve evolution, depth by gradient descent, and 3-D motion by least squares within each region of segmentation. Curve evolution is implemented via level sets for topology independence and numerical stability. This algorithm and its implementation are verified on synthetic and real image sequences. Viewers presented with anaglyphs of stereoscopic images constructed from the algorithm's output reported a strong perception of depth.

Algorithms↗
PubMed2006

Subpixel estimation of shifts directly in the Fourier domain.

In this paper, we establish the exact relationship between the continuous and the discrete phase difference of two shifted images, and show that their discrete phase difference is a two-dimensional sawtooth signal. Subpixel registration can, thus, be performed directly in the Fourier domain by counting number of cycles of the phase difference matrix along each frequency axis. The subpixel portion is given by the noninteger fraction of the last cycle along each axis. The problem is formulated as an overdetermined homogeneous quadratic cost function under rank constraint for the phase difference, and the shape constraint for the filter that computes the group delay. The optimal tradeoff for imposing the constraints is determined using the method of generalized cross validation. Also, in order to robustify the solution, we assume a mixture model of inlying and outlying estimated shifts and truncate our quadratic cost function using expectation maximization.

Algorithms↗
PubMed2006

Multidimensional multichannel FIR deconvolution using Gröbner bases.

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gröbner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gröbner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gröbner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.

Algorithms↗
PubMed2002

Generation of anisotropic-smoothness regularization filters for EIT.

In the inverse conductivity problem, as in any ill-posed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper.

Algorithms↗
PubMed2004

Adaptive probabilistic neural networks for pattern classification in time-varying environment.

In this paper, we propose a new class of probabilistic neural networks (PNNs) working in nonstationary environment. The novelty is summarized as follows: 1) We formulate the problem of pattern classification in nonstationary environment as the prediction problem and design a probabilistic neural network to classify patterns having time-varying probability distributions. We note that the problem of pattern classification in the nonstationary case is closely connected with the problem of prediction because on the basis of a learning sequence of the length n, a pattern in the moment n + k, k > or = 1 should be classified. 2) We present, for the first time in literature, definitions of optimality of PNNs in time-varying environment. Moreover, we prove that our PNNs asymptotically approach the Bayes-optimal (time-varying) decision surface. 3) We investigate the speed of convergence of constructed PNNs. 4) We design in detail PNNs based on Parzen kernels and multivariate Hermite series.

Algorithms↗
PubMed2005

Posterior probability support vector machines for unbalanced data.

This paper proposes a complete framework of posterior probability support vector machines (PPSVMs) for weighted training samples using modified concepts of risks, linear separability, margin, and optimal hyperplane. Within this framework, a new optimization problem for unbalanced classification problems is formulated and a new concept of support vectors established. Furthermore, a soft PPSVM with an interpretable parameter v is obtained which is similar to the v-SVM developed by Schölkopf et al., and an empirical method for determining the posterior probability is proposed as a new approach to determine v. The main advantage of an PPSVM classifier lies in that fact that it is closer to the Bayes optimal without knowing the distributions. To validate the proposed method, two synthetic classification examples are used to illustrate the logical correctness of PPSVMs and their relationship to regular SVMs and Bayesian methods. Several other classification experiments are conducted to demonstrate that the performance of PPSVMs is better than regular SVMs in some cases. Compared with fuzzy support vector machines (FSVMs), the proposed PPSVM is a natural and an analytical extension of regular SVMs based on the statistical learning theory.

Algorithms↗